arXiv subjects

Sombuddha Bhattacharyya

Publications and source records attributed to Sombuddha Bhattacharyya.

16 recordsLinked to original sources

arXiv2026

Direct and inverse problem for bi-wave equation with time-dependent coefficients from partial data

In this article, we study a direct and an inverse problem for the bi-wave operator $(\Box^2)$ along with second and lower order time-dependent perturbations. In the direct problem, we prove that the operator is well-posed, given initial and boundary data in suitable function spaces. In the inverse problem, we prove uniqueness of the lower order time-dependent perturbations from the partial input-output operator. The restriction in the measurements are considered by restricting some of the Neumann data over a portion of the lateral boundary.

math.AP↗

arXiv2025

The Momentum Light Ray Transform

In this article, we study Momentum Light Ray Transform (MLRT) on symmetric tensor fields. MLRT is an integral transform in time-space domain ($(t,x)\in \mathbb{R}^{1+n}$), which integrates a scalar function or a tensor field along the light rays with a polynomial type weight. We explore necessary and sufficient conditions for injectivity of MLRT, over general order tensors on space dimension $\geq 2$, from full and restricted measurements. Furthermore, we develop an inversion algorithm for MLRTs in the restricted measurement setting. To prove the results, we use tools from tensor tomography, geometry, and analysis.

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arXiv2025

Density results of biharmonic functions on symmetric tensor fields and their applications to inverse problems

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric tensor fields up to order three, in a bounded domain. Furthermore, as an application of the density results, in dimension two or higher, we solve a partial data inverse problem for a biharmonic operator with nonlinear anisotropic third and lower order perturbations. For the inverse problem, we take the Dirichlet data to be supported in an arbitrarily small open set of the boundary and measure the Neumann data on the same set. Note that the analogous problem for linear perturbations are still unknown. So far, partial data problems recovering nonlinear perturbations were studied only up to vector fields. The full data analogues of the inverse problem has recently been studied for three or higher dimensions.

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arXiv2024

Recovery of piecewise smooth parameters in an acoustic-gravitational system of equations from exterior Cauchy data

In this paper, we study an inverse problem for an acoustic-gravitational system whose principal symbol is identical to that of an acoustic wave operator. The displacement vector of a gas or liquid between the unperturbed and perturbed flow is denoted by $u(t,x)$. It satisfies a partial differential equation (PDE) system with a principal symbol corresponding to an acoustic wave operator, but with additional terms to account for a global gravitational field and self-gravitation. These factors make the operator nonlocal, as it depends on the wave speed and density of mass. We assume that all parameters are piecewise smooth in $\mathbb{R}^3$ (i.e., smooth everywhere except for jump discontinuities across closed hypersurfaces called interfaces) but unknown inside a bounded domain $Ω$. We are given the solution operator for this acoustic-gravitational system, but only outside $Ω$ and only for initial data supported outside $Ω$. Using high-frequency waves, we prove that the piecewise smooth wave speed and density are uniquely determined by this map under certain geometric conditions.

math.AP↗

arXiv2024

Local data inverse problem for the polyharmonic operator with anisotropic perturbations

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

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arXiv2023

Inverse Problems For Third-Order Nonlinear Perturbations Of Biharmonic Operators

We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in $\mathbb{R}^n$, where $n\geq 3$. By imposing appropriate assumptions on the nonlinearity, we demonstrate that the Dirichlet-to-Neumann map, known on the boundary of the domain, uniquely determines the genuinely nonlinear tensorial third-order perturbations of the biharmonic operator. The proof relies on the inversion of certain generalized momentum ray transforms on symmetric tensor fields. Notably, the corresponding inverse boundary problem for linear tensorial third-order perturbations of the biharmonic operator remains an open question.

math.AP↗

arXiv2022

Recovery of piecewise smooth density and Lamé parameters from high-frequency exterior Cauchy data

We consider an isotropic elastic medium occupying a bounded domain D whose density and Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside the domain D and only for initial data supported outside D, and we study the recovery of the density and Lamé parameters. For known density, results have recently been obtained using the scattering control method to recover wave speeds. Here, we extend this result to include the recovery of the density in addition to the Lamé parameters under certain geometric conditions using techniques from microlocal analysis and a connection to local tensor tomography.

math.AP↗

arXiv2022

Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators

We revisit the problem of recovering wave speeds and density across a curved interface from reflected wave amplitudes. Such amplitudes have been exploited for decades in (exploration) seismology in this context. However, the analysis in seismology has been based on linearization and mostly flat interfaces. Here, we present a nonlinear analysis allowing curved interfaces, establish uniqueness and provide a reconstruction, while making the notion of amplitude precise through a procedure rooted in microlocal analysis.

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arXiv2021

Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data

We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized MRT. The uniqueness result and the inversion formula we prove for generalized MRT could be of independent interest and we expect it to be applicable to other inverse problems for higher order operators involving tensor perturbations.

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arXiv2021

An Inverse Problem on Determining Second Order Symmetric Tensor for Perturbed Biharmonic Operator

This article offers a study of the Calderón type inverse problem of determining up to second order coefficients of the higher order elliptic operator. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain.

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arXiv2021

Inverse Problem for Kirchhoff-Love Plate Equation

We consider the two-dimensional Kirchhoff-Love plate equation in the context of elasticity modeling the stresses and deformations in thin plates subjected to forces and moments. We establish global recovery of the material parameters like bending stiffness, Poisson coefficient, Lamé parameters from the associated boundary Cauchy data of the equation.

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arXiv2020

Inverse Problem for Fractional-Laplacian with Lower Order Non-local Perturbations

In this article, we study a model problem featuring a Lévy process in a domain with semi-transparent boundary by considering the following perturbed fractional Laplacian operator \[\mathscr{L}_{b,q} := (-Δ)^t + (-Δ)_Ω^{s/2} \ b (-Δ)_Ω^{s/2} + q, \quad 0<s<t<1\] on a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$. While the non-locality of the fraction Laplacian $(-Δ)^t$ depends on entire $\mathbb{R}^n$, in its non-local perturbation the non-locality depends on the domain $Ω$ through the regional fractional Laplacian term $(-Δ)^{s/2}_Ω$ and $b$ exhibits the semi-transparency of the process. We analyze the well-posedness of the model and certain qualitative property like unique continuation property, Runge approximation scheme considering its regional non-local perturbation. Then we move into studying the inverse problem and find that by knowing the corresponding Dirichlet to Neumann map (D-N map) of $\mathscr{L}_{b,c}$ on the exterior domain $\mathbb{R}^n \setminus Ω$, it is possible to determine the lower order perturbations `$b$',`$q$' in $Ω$. We also discuss the recovery of `$b$', `$q$' from a single measurement and its limitations.

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arXiv2018

An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrödinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.

math.AP↗

arXiv2018

Local Uniqueness of The Density From Partial Boundary Data for Isotropic Elastodynamics

We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3-dimensional object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary.

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arXiv2018

Inverse boundary value problem of determining up to second order tensors appear in the lower order perturbations of the polyharmonic operator

We consider the following perturbed polyharmonic operator $\Lc(x,D)$ of order $2m$ defined in a bounded domain $Ω\subset \mathbb{R}^n, n\geq 3$ with smooth boundary, as \begin{equation*} \Lc(x,D) \equiv (-Δ)^m + \sum_{j,k=1}^{n}A_{jk} D_{j}D_{k} + \sum_{j=1}^{n}B_{j} D_{j} + q(x), \end{equation*} where $A$ is a symmetric $2$-tensor field, $B$ and $q$ are vector field and scalar potential respectively. We show that the coefficients $A=[A_{jk}]$, $B=(B_j)$ and $q$ can be recovered from the associated Dirichlet-to-Neumann data on the boundary. Note that, this result shows an example of determining higher order ($2$nd order) symmetric tensor field in the class of inverse boundary value problem.

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arXiv2017

Optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data

We consider the inverse boundary value problem for operators of the form $-\triangle+q$ in an infinite domain $Ω=\mathbb{R}\timesω\subset\mathbb{R}^{1+n}$, $n\geq3$, with a periodic potential $q$. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form $Γ_1=\mathbb{R}\timesγ_1$, with $γ_1$ being the complement either of a flat or spherical portion of $\partialω$, we prove that a log-type stability estimate holds.

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